How to Find the Derivative of csc(x) Using the Quotient Rule

Derivative of: csc(x)

The derivative of csc(x) can be found by using the quotient rule

The derivative of csc(x) can be found by using the quotient rule. Let’s start by expressing csc(x) as 1/sin(x):

csc(x) = 1/sin(x)

Next, we can rewrite this expression as:

csc(x) = (1/sin(x)) * (sin(x)/sin(x)) = sin(x)/sin^2(x)

Now let’s find the derivative of sin(x)/sin^2(x) using the quotient rule.

The quotient rule states that if we have a function f(x) = g(x)/h(x), then the derivative can be found using the formula:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / h(x)^2

In our case, g(x) = sin(x) and h(x) = sin^2(x).

Now, let’s find the derivatives of g(x) and h(x):

g'(x) = cos(x) (derivative of sin(x))
h'(x) = 2sin(x)cos(x) (derivative of sin^2(x) using the chain rule)

Substituting these derivatives into the quotient rule formula, we get:

f'(x) = (cos(x) * sin^2(x) – sin(x) * 2sin(x)cos(x)) / sin^4(x)

Simplifying this expression further, we obtain:

f'(x) = (cos(x) * sin^2(x) – 2sin^2(x)cos(x)) / sin^4(x)

Factorizing cos(x) from the numerator, we have:

f'(x) = [cos(x) * (sin^2(x) – 2sin^2(x))] / sin^4(x)

Combining like terms, we get:

f'(x) = -sin^2(x) / sin^4(x)

Finally, we can simplify by using the identity sin^2(x) = 1 – cos^2(x):

f'(x) = -(1 – cos^2(x))/sin^4(x)

Therefore, the derivative of csc(x) with respect to x is:

d/dx (csc(x)) = -(1 – cos^2(x))/sin^4(x)

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